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Re: This Week’s Finds in Mathematical Physics (Week 246)

First of all, the application of mathematical techniques from string theory to the problem of heavy ion collisions is completely separate from the issue of whether string theory is a correct theory of fundamental particle physics. The stuff you’re talking about is just a clever new way of doing calculations in Standard Model physics. Even if it works, it doesn’t mean the universe is made of strings.

Do you know this? I ask because I just realized, to my horror, that it might not be obvious to layfolk! It’s obvious to physicists.

It’s as if Einstein figured out a way to use math from general relativity to solve problems in hydrodynamics. Suppose it turned out that these methods could correctly predict what happens when you flush your toilet. This would not mean general relativity is correct! To test general relativity you need to look at bending starlight or black holes, not flush toilets.

Second of all, it’s not clear how well these AdS/CFT methods actually work. Since string theory is supersymmetric, these methods actually apply to something called N=4N = 4 supersymmetric Yang–Mills theory. This is similar to the ordinary Yang–Mills theory that we use to describe quarks and gluons… but it’s different. So, it only gives approximately correct answers for the real-world problems involving quarks and gluons, and there’s a bit debate over how well it works, and how much it’s been hyped.

Over at that blog you mentioned, Backreaction, you’ll see that Larry McLerran has given Brian Greene a “Pinocchio award” for overstating how well these AdS/CFT methods work for relativistic heavy ion collisions! Here’s one of his slides:

Here ‘N=4N = 4 SUSY Yang–Mills’ is the theory that string theory techniques can be applied to, while quantum chromodynamics (= ‘QCD’) is the theory that actually describes the strongly coupled quark-gluon plasma (= ‘sQGP’) they’re seeing at the Relativistic Heavy Ion Collider (= ‘RHIC’). The slide is pointing out how these are quite different.

Re: This Week’s Finds in Mathematical Physics (Week 246)

John Baez wrote:

The stuff you’re talking about is just a clever new way of doing calculations in Standard Model physics. Even if it works, it doesn’t mean the universe is made of strings.

When my lawyer and artist friends ask me (“that physics guy”) about string theory, this is one of the points I try to get across, and it seems to transmit well. It’s just string theory going back to its roots, after all. So, leaving aside the point raised by Polchinski, the idea appears to be pretty easily grokkable.

Polchinski wrote:

String-theory skeptics could take the point of view that it is just a mathematical spinoff. However, one of the repeated lessons of physics is unity — nature uses a small number of principles in diverse ways. And so the quantum gravity that is manifesting itself in dual form at Brookhaven is likely to be the same one that operates everywhere else in the universe.

One could easily take this idea too far. In undergrad quantum mechanics, I was taught to solve the hydrogen atom with a method which is essentially the grandchild of superstrings, but that doesn’t mean the universe is “stringy”!

My biggest concern with the AdS/CFT business is not that it shows string theory is correct, or anything like that. Instead, it worries me that a discussion which purports to concern the sociology of science seems to sidestep the question of what a significant fraction of the scientists are actually doing. If a large group of people are not being seduced by the “Theory of Everything” grail-shaped beacon, but instead choosing to work in a mathematically related field with direct ties to experiment, then doesn’t that have incredible importance for the psychological and sociological parts of the argument? This holds true, I think, even if the QGP calculations never really bear fruit — say, if the whole thing doesn’t give much more precise answers than dimensional analysis.

To steal the Vegas analogy, it’s as if a group of gamblers had decided to play the odds a better way: they put computers in their shoes to predict where roulette balls will fall, or they make side bets with people around the craps table who have superstitious ideas about lucky numbers. They’re not playing the same game as the tourists, but they’re seeing real money. Can any study of the gambling world rightfully ignore them?

Conflict of Interest Disclaimer: my only stake in the String Wars is a small one. I took Barton Zwiebach’s String Theory for Undergraduates (8.251) and helped proofread the textbook, when it was only a stack of LaTeX documents. I worked the exercises in the last few chapters to make sure that an actual undergraduate with no intellectual superpowers had a chance of solving them. In my mind, this was a good thing to do even if the whole thing goes kaput and the M in M-theory turns out to stand for “mud pie”. After all, we’ll only be able to tell if the ideas are good or not if enough brains can gather around them. CITOKATE — “criticism is the only known antidote to error”.

Re: This Week’s Finds in Mathematical Physics (Week 246)

I’m all in favor of string theorists using the technology they’ve developed to tackle real-world problems like the study of quark-gluon plasma. I just wanted to make sure Mr. Serg271 here understood the difference between the supersymetric quark-gluon plasmas these folks are studying, the actual quark-gluon plasmas folks are creating at Brookhaven, and string theory as a theory of fundamental physics.

I’m not surprised that a bunch of string theorists, starved for contact with experiment, would enjoy working on this stuff. Grail-shaped beacons are all very well and good, but physicists only bring home the bacon when they predict the results of experiment.

So, the big question is: how much reliable information can we obtain about real-world quark-gluon plasmas from studying their supersymmetric analogues? I’d like to know… but I guess this is very controversial, since I’ve seen diametrically opposite claims.

I would also enjoy knowing how many string theorists are working on this stuff. Clifford Johnson claims it’s “a huge percentage”. Any idea what percentage that is, or how many people it amounts to?

I’m less interested in grinding some sociological axe than getting some data.
I’m sick of the String Wars — I only wrote about these books because I felt some kind of duty to do so. What I really want to talk about is Schur functors, Littlewood–Richardson rules, cohomology of Grassmannians, and groupoidification! But that’ll be next Week’s Finds.

Re: This Week’s Finds in Mathematical Physics (Week 246)

Maybe I should try throwing together a script which browses the arXiv and counts how many different authors have written papers relating to or citing a given publication (in this case, perhaps hep-ph/0608177). Heck, I could probably get a journal article in social networks out of that.

Re: This Week’s Finds in Mathematical Physics (Week 246)

Re: This Week’s Finds in Mathematical Physics (Week 246)

Well that I definitely understand. I also got the idea that “stringy” SUSY QGC is not exactly the same as that of Standard Model. But I had heard the opinion that the prediction of SUSY QGC seems an “unexpectedly good” fit to experiment. I guess we have to wait for more results from RHIC or LHC. But if this correspondence works, isn’t it an argument in favor of string theory? At least that does mean it’s not self-contradictory and not trivial (in the sense that it potentially can predict something).

Re: This Week’s Finds in Mathematical Physics (Week 246)

I imagine that the success of SUSY QCD depends on which quantity you are looking at. The susceptibility seems only to be off by a few percent, whereas the beta-function is off by a factor infinity, since N=4 SYM remains scale invariant after quantization.

The problem with AdS/CFT in this context is that it is an uncontrolled approximation, AFAIU. That it involves infinitely many colors is not a problem, since you typically can calculate n-color corrections as a power series in 1/n, and 1/3 is close to 1/infinity.

In contrast, a theory with SUSY, especially four SUSIES, is qualitatively different from a theory without SUSY. I have at least never heard of somebody considering N = 4-ε SYM, work out the corrections as a power series in ε, and set ε = 4 in the end, which one would expect to do if one could turn N=4 SYM into a starting point for a controlled approximation.

Re: This Week’s Finds in Mathematical Physics (Week 246)

In TWF 246, John wrote:

Once I drove through Las Vegas, where there really is just one game in town: gambling. I stopped and took a look. I saw the big fancy casinos. I saw the glazed-eyed grannies feeding quarters into slot machines, hoping to strike it rich someday. It was clear: the odds were stacked against me. But, I didn’t respond by saying “Oh well - it’s the only game in town” and starting to play.

Instead, I left that town.

Earlier on this blog page, he also wrote:

It’s as if Einstein figured out a way to use math from general relativity to solve problems in hydrodynamics. Suppose it turned out that these methods could correctly predict what happens when you flush your toilet. This would not mean general relativity is correct! To test general relativity you need to look at bending starlight or black holes, not flush toilets.

I laughed out loud when I read these. Arguments about the “String Wars” aside, I feel that these two passages demonstrate John’s amazing ability for a turn of phrase! He gets the Oscar “Lifetime Achievement Award for Explaining the Complicated in Simple Terms.”

Re: This Week’s Finds in Mathematical Physics (Week 246)

It is true that N=4 and QCD are quite different. However, AdS/CFT can be deformed to cases which are non-SUSY and or non-conformal, and many view this as an existence proof that a string theory dual to QCD is possible. I really yearn for it because it will put to rest a lot of absurd criticisms of string theory. For example:

“Even if it works, it doesn’t mean the universe is made of strings.”

I imagine a conversation like this many years ago.

“Even if this so-called `wave-particle duality’ you propose works, it doesn’t mean the particles in our universe are actually waves. It’s just some new-fangled mathematical mumbo-jumbo.”

Gauge/gravity duality is no different. If you prefer to say the world is “made of gluons”, which in a certain limit behave like coherent states of gravitons, that’s fine. If I prefer to say the world is “made of gravitons, which move around in some higher dimensional curved space”, but in certain limits behave like coherent states of gluons in 4-dimensional Minkowski space, that’s equally fine. Each description is useful in different regimes, but none is more “true” than the other.

To put it another way: there are hundreds or thousands of people around the world who are working on QCD. In fact, they are working on string theory; they just don’t know it yet.

Re: This Week’s Finds in Mathematical Physics (Week 246)

It is true that N=4 and QCD are quite different. However, AdS/CFT can be deformed to cases which are non-SUSY and or non-conformal, and many view this as an existence proof that a string theory dual to QCD is possible.

Alas, for this to be useful the putative string dual must be tractable. If you can replace physical QCD with a much more complicated theory, you haven’t gained anything.

“Even if it works, it doesn’t mean the universe is made of strings.”

I imagine a conversation like this many years ago.

“Even if this so-called `wave-particle duality’ you propose works, it doesn’t mean the particles in our universe are actually waves. It’s just some new-fangled mathematical mumbo-jumbo.”

Another conversation from long ago:

“Even if this ether theory that you propose works, it does not mean that electromagnetic waves in our universe are really waves in the ether.”

Anyway, if quantum gravity combines background independence with locality, QJT is the only game in town. This is because only QJT supports the 4D diff anomalies which are necessary to have correlators depend on separation. As is well known, even infinite conformal symmetry in the strict sense is incompatible with locality. Nontrivial correlators, i.e. a positive anomalous dimension, requires an anomaly, and diffeomorphisms work the same way as conformal transformations.

Re: This Week’s Finds in Mathematical Physics (Week 246)

“Each description is useful in different regimes, but none is more “true” than the other”

The description of QCD in terms of gauge fields is certainly more “true” than any known description in terms of strings. QCD is a fully non-perturbative theory, unlike string theory. Not only has no one yet found a version of string theory that accurately approximates QCD at weak (string) coupling, but there is not even a proposal for a non-perturbative string theory that would be fully equivalent to QCD.

And, being wildly optimistic and assuming you find such a thing, the universe will still not be “made of strings”, since there is the rest of the standard model to take into account. You have to find a non-perturbative string theory whose strong-coupling limit gives you electroweak gauge fields, spinor fields, the Higgs mechanism, etc. Or else you have to get this out of weakly coupled strings/branes, an idea which has led to the “landscape” and pretty conclusive failure.

Comparing the current situation of string theory vs. QFT to wave-particle duality doesn’t seem to me to hold water. When people were talking about wave-particle duality they had a specific, testable and validated model to point to.

Working on finding a string theory dual to QCD is certainly a valid project with some promise. But it doesn’t justify in any way claiming vindication for the project of unifying quantum gravity and the Standard Model using 10d strings.

atical Physics (Week 246)

If we end up with a coherent and consistent unified theory of the universe, involving extremely complicated mathematics, do we believe that this represents “reality”?

If all the rich mathematics springs from a simple principle then, yes, I would be inclined to do so.

Or, better, put the other way around: I would be surprised, then, if all that rich structure had no place in the reality we perceive.

The math used in string theory may be complicated and demanding. But so is that of the 3-body problem in Newtonian mechanics.

The principle from which all this math springs from is, however, rather simple: pass from the functional
γ↦e−m∫γds
\gamma \mapsto
e^{-m\int_\gamma d s}
used in quantum field theory
on maps γ:[0,1]→M\gamma : [0,1] \to M from the interval into some pseudo-Riemannian space

(1)(M,g)
(M,g)

to the functional

(2)Σ↦e−T2∫Σd2s
\Sigma \mapsto
e^{-\frac{T}{2}\int_\Sigma d^2s}

on maps from 2-dimensional spaces into target space.

Quantizing (and second quantizing) this gives all the rich structure that is called “string theory”.

And all the indeterminacy: what would be more irritating: if the formula (2) completely encoded the mass of the pion, or if it did not?

The remarkable thing is that we can choose target spaces (1) such that (2) knows about anything like pions at all.

If string theory turns out to have nothing to do with physics, it will remain a pool of rich mathematics.

The dynamics of string backgrounds is that of Ricci flow (or the other way around).

If you like to put it that way: points in that infamous “string landscape” are fixed points of a generalized Ricci flow.

I’d call that “rich mathematics”. Though it is certainly complicated, too.

There are mathematicians working on a mathematical field called topological T-duality who don’t know the first thing about string theory. Their field originates in string theory and is being pursued as a mathematical entity in its own right.

As you know, the latest in that direction is geom. Langlands. Rich and complicated.

Re: This Weeks Finds in Mathematical Physics (Week 246)

So Atiyah’s comment is rather odd from your perspective? Necessarily, any simply principled ‘theory of everything’ will require complicated techniques to extract the way our messy universe is. Simple consequences of simple principles would be too, well, simple?

Re: This Week’s Finds in Mathematical Physics (Week 246)

Urs and David,

I actually sent Atiyah a draft of the book, in particular to ask him about whether I was accurately reflecting his opinions in the final section where I discussed some of what he had said at the recent conference in honor of Gelfand. He sent me back some comments and a draft of the writeup for his talk that I ended up quoting in the book.

Atiyah is definitely more of a fan of string/M-theory than I am, and he reminded me that he is a co-author with Witten of a paper on M-theory and has an extremely high opinion of Witten’s judgement in these matters. I don’t want to put words in his mouth, but I think what he wrote for the Gelfand conference speaks for itself. I believe he sees string/M-theory as a very fruitful source of mathematical ideas and something that has probably captured some aspect of physical reality, but he’s no fan of the complicated mess that 10/11 dimensions leads one into. The fact that this complicated mess invokes algebraic geometry of 3-folds, K-theory, index theory, the Ricci flow and all sorts of other sophisticated mathematical technology is not necessarily something he would see as a positive thing. Atiyah knows those subjects well enough to distinguish between a deep and a superficial use of them, and I think he’s fairly explicit in saying this is not a deep use of mathematics.

I think his attitude is not fundamentally different than that of David Gross, who continually makes the point that we “don’t know what string theory is”, that the current understanding of string theory both lacks any deep new symmetry principle and a non-perturbative formulation. Gross hope that a deeper understanding of string theory will lead to a revision of our ideas about space and time. This is pretty much the same as what I think Atiyah would like to see, a mathematically deeper and more geometrical insight into what is going on with string theory, one that would do away with the rather complicated and ugly constructions it currently uses to (unsuccessfully) connect to reality.

And, by the way, that “unsuccessfully” is the point. If these constructions actually led to any accurate predictions of anything about the world, Gross or Atiyah wouldn’t be going on so much about how important it is to look for something simpler.

Re: This Week’s Finds in Mathematical Physics (Week 246)

So, then, it seems that when Atiyah refers to (in the above quote) “extremely complicated mathematics”, he is actually referring to the infamous algorithmic complexity (meaning: it takes us lots of pages to define) of those backgrounds that are known to produce something close to the standard model.

If that’s what is meant, I would just remark that I would maybe hesitate to call this kind of complexity “complicated mathematics”. It is rather “complicated data”:

some fixed points of your Ricci flow will take you more effort (more parameters) to specify than others. The math governing them is the same in both cases.

Or: some solutions of Newton’s equations may require specifying more data – like for instance that of the solution describing the 9-body problem being our solar system – than others – like that describing just the earth-moon system.

Once some people were concerned about this algorithmic complexity of the solar system. At least a great mind like Kepler was. Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids. #.

“Oh how naive!”, we say today. We know that the precise distances and masses in the solar system are a result of various arbitrary coincidences in the details of the history of its formation, and that there is no reason to expect that the algorithmic complexity of this particular solution of Newton’s equations is less than that given by specifying all these parameters one by one.

But somehow, we are in a similar situation now with respect to the standard model of particle physics as Kepler was back then with respect to the solar system.

We do not know: are the number and masses and couplings of the particles in the standard model a result of various arbitrary coincidences in the details of the history of its formation? Or should we expect their algorithmic complexity to be much less?

Re: This Weeks Finds in Mathematical Physics (Week 246)

(…)it will remain a pool of rich mathematics.

Is mathematics endless? Is it limited by our own mind? Or is it external to our mind? (If these questions make sense)… What could the existence of a richer and richer (or more complicated, for what is worth) mathematics tells us about reality itself?

Re: This Week’s Finds in Mathematical Physics (Week 246)

I think the expression “pool of rich mathematics” can only be taken here as a (subjective) statement made by someone relying on his/her past experience rather than a claim that “math is endless” as you ask (which I think isn’t).

About your other questions, complicated mathematics may not have anything to do with reality, just like it’s possible to imagine what the life of a bunch of blue elephants who speak seven languages would be like: pretty complicated but not real. Since maths is always purely imagined (in the sense of not exerting any influence on non-humans, like a cat or a rock) the same non-relevance argument applies.

Re: This Week’s Finds in Mathematical Physics (Week 246)

complicated mathematics may not have anything to do with reality, just like it’s possible to imagine what the life of a bunch of blue elephants who speak seven languages would be like: pretty complicated but not real

No. That is not the line of reasoning that I meant with my question. The fact that we can do complicated mathematics has implications on how our mind works. Why is our intellect mathematically driven and what clues does this fact give about reality? If mathematics is in principle an ever evolving activity as far as we humans evolve, why does reality allow such a state of affairs? Why is it not much more constrained? Or is it really constrained (but we still do not know how far)? Then, why?

Feynman; Re: This Weeks Finds in Mathematical Physics (Week 246)

Is Math endless? Probably. Is math about the Universe endless? Possible, but less probable, in the following sense.

Richard Feynman, most of the time, exulted in how simple assumptions could lead to robustly complicated behaviors. He loved solving problems on the fly, and, in his famous course “Physics X”, even liked to solve audience-supplied problems in public. Well, in a classroom of grad students, a few undergrads, and visitors.

Yet once in a while, he entertained an alternative view. He discussed with me more than once (1968 until a few months before his death) the possibility that there are actually an infinite number of “natural laws”, each expressed by its own equations, with no more fundamental meta-equation. Perhaps, he speculated, some of these infinite number of natural laws only occur at very high energies, or very weird combinations of parameters, or at different ages of the universe.

He always claimed that there were plenty of people, even at Caltech, who were better at Math than him. He gave some priority to his “physical intuition.” He even told me that he tolerated my relative weakness in Math because I did have good physical intuition. He was skeptical of String Theory as perhaps pretty Math, but not established to have any connection to Physics as he saw it.

Re: This Week’s Finds in Mathematical Physics (Week 246)

For example, some people have tried to refute the claim that string theory makes no testable predictions by arguing that it predicts the existence of gravity! This is better known as a “retrodiction”.

This is not even retrodiction! Retrodiction is (in the English Wikipedia’s current words) ‘the act of making a prediction about the past’. Logically, it is still a scientific prediction (that is, a factual claim whose truth we do not know but which we believe and intend to test). Retrodiction may be used, for example, in archaeology, to predict what the contents of a site will be before it is excavated.

Rather, ‘predicting’ gravity is more akin to postdiction, which is (to edit the text of Wikipedia) ‘an effect of hindsight bias that explains claimed predictions of significant events, such as plane crashes and natural disasters’. This term is used by those sceptical of paranormal phenonema, properly extended here to scepticism of string theory.

Indeed, in fundamental physics that seeks to describe the nature of time itself, the difference between prediction and retrodiction is ill defined, while postdiction is quite different. The difference between (scientific) pre-/retro-diction and (pseudoscientific) postdiction is the position in the subjective timeline of the one making the -diction.

But, the “only game in town” argument is still flawed.

I certainly hope that nobody is seriously using this phrase as an argument! It was originally the punch line to a sarcastic joke, quoted here from an article about online casinos:

Reminds me of one of the more legendary gamblers of all time named Canada Bill. His gambling immortality does not rest on his gambling prowess, nor his formidable wins or losses. He is remembered by a single line he once uttered on the Mississippi, a phrase recited by a myriad of gamblers since. Bill was losing his entire bankroll at Faro when a friend approached and said, “Bill, don’t you know this game is crooked?” “Yes,” answered Canada Bill, “but it’s the only game in town.”

Your Las Vegas parable is just a less sarcastic way of pointing out precisely the same point. Anybody earnestly using this phrase is making a fool of themself (like an anti-immigration politician citing Robert Frost).

Re: This Week’s Finds in Mathematical Physics (Week 246)

Many historians of physics reckoned that general relativity’s accounting for the anomalous precession of Mercury’s perihelion was hugely influential in getting physicists to sign up to it. Now what kind of ‘-diction’ was involved there? The anomaly had been known for several years.

Philosophers of science usually call it a retrodiction. Fierce debate broke out perhaps twenty years ago about the degree of support to a theory in such a case. Did it matter whether the data had been used in the construction of the theory, etc. Indeed, Einstein had rejected an earlier theory because it was incompatible with this data.

Some argue that it’s not the temporal order of the devising of a theory and the observation of data that matters, but rather a question of the extent to which to explain the data you need to fix certain parameters, i.e., that there is a timeless relation of support between evidence and theory.

When you look into the details of a case, things become mighty complicated.

Re: This Week’s Finds in Mathematical Physics (Week 246)

Some argue that it’s not the temporal order of the devising of a theory and the observation of data that matters, but rather a question of the extent to which to explain the data you need to fix certain parameters, i.e., that there is a timeless relation of support between evidence and theory.

Yes, exactly.

The Schrödinger equation (plus some extra data) still predicts the diameter of the hydrogen atom, doesn’t it?

We don’t teach students that Schrödinger’s equation “retrodicts” or “postdicts” the size of the hydrogen atom, just because the comparison with experiment had been done long time ago.

I would not think that the term “predictions of a theory” is usually used in the sense of “gives us a vision of the future” (though that may be a special case), but in the sense of “these facts are derivable from the axioms of the theory plus given extra data”.

In math we say “axioms” and “implications/theorems/corollaries”. In physics we say “theory” and “predictions”.

The fight about whether string theory pre- posts- or retrodicts something is hence like many of those fights in the String Wars: it is not so much about the theory itself, but about its sociological consequences.

Technically I think it is quite right to say that string theory predicts gravity. But when the discussion is all about whether or not we, as a society, are taking great risks by spending so many resources on this theory, we may tend to dislike stating this technically correct statement this way, because we may feel that it does not sufficiently amplify the point that this prediction is rather useless, for our practical purposes.

Re: This Week’s Finds in Mathematical Physics (Week 246)

Of course, one advantage of a future prediction is that without knowing the details of the predicting theory, if the prediction is surprising and it turns out true, then you will be impressed.

On the other hand, in the case of retrodiction, without diving into the details of the theory, as far as you know the theorists might have just tweaked the parameter knobs of their theory to get the right result.

Re: This Week’s Finds in Mathematical Physics (Week 246)

How many “knobs” are there which we can tweak to make gravity not emerge? Closed strings obeying special relativity yield graviton states upon quantization. I can change the string length, or I can ramp up the dilaton expectation value to vary the coupling, but just how hard is it to make the basic form of gravitation go away?

Re: This Week’s Finds in Mathematical Physics (Week 246)

In math we say “axioms” and “implications/theorems/corollaries”. In physics we say “theory” and “predictions”.

Suppose someone comes up with a theory of physics. We can imagine three things happening:

The theory lets us to calculate some quantity whose value we didn’t previously know. We measure the quantity and the theory turns out to be right.

The theory lets us calculate some quantity whose value we previously knew, but could not calculate using previous theories.

The theory lets us calculate some quantity whose value we previously knew, and could calculate using previous theories.

Very roughly speaking (see the fine print below), we get really excited when a theory does 1. We get a bit excited when a theory does 2. And, we feel the theory isn’t obviously wrong when it does 3.

For example:

General relativity reduces to Newtonian gravity in a suitable limit — that’s an event of type 3. It told Einstein that general relativity isn’t obviously wrong.

General relativity predicted the rate at which Mercury’s orbit precesses — that’s an event of type 2. This got Einstein and other people a bit excited about general relativity.

But also, general relativity predicted how much starlight bends when it goes around the sun — that’s an event of type 1. This got Einstein and other people really excited about general relativity. This is when Einstein made the front page of the New York Times, with a headline reading:

Lights All Askew In The Heavens

Men Of Science More Or Less Agog Over Results Of Eclipse Observations

Einstein Theory Triumphs

When I spoke of ‘predictions’ in This Week’s Finds, I was referring to events of type 1 and (to a lesser extent) type 2. The ‘prediction’ of gravity by string theory is an event of type 3.

The fight about whether string theory pre- posts- or retrodicts something is hence like many of those fights in the String Wars: it is not so much about the theory itself, but about its sociological consequences.

I don’t think the difference between events of type 1, 2 and 3 is merely ‘sociological’. I really think a good scientist should — other things being equal — be more excited by events of type 1 than by events of type 2, and more excited by events of type 2 than events of type 3. There are good reasons for this: the low-numbered events really do have more ‘confirmatory power’, all else being equal. Consult your local philosopher of science (David) for further discussion of ‘confirmatory power’.

Of course, other things aren’t always equal. We know the masses of elementary particles already, so if someone invents a theory that lets us calculate them all, it will be an event of type 2 — but if the theory is very nice, we’ll get really excited, because people have tried and failed to do this for so long!

Etcetera: one can imagine all sorts of scenarios where we’re bored stiff by an event of type 1, or fantastically thrilled by an event of type 3.

Nonetheless, I still think there’s something to my point.

If an event of type 1 occurs for string theory, there will be a headline on the front page of the New York Times about it, and an issue of This Week’s Finds specially devoted to it!

Re: This Week’s Finds in Mathematical Physics (Week 246)

Etcetera: one can imagine all sorts of scenarios where we’re bored stiff by an event of type 1, or fantastically thrilled by an event of type 3.

Case in point:

I seem to have lent my copy of Intellectual Impostures to a friend, so I’d have to look up the references from scratch, but I recall that Weinberg among others has pointed out that the “type 1” prediction of GR, the deflection of light during a solar eclipse, was in fact a worse test than the perihelion of Mercury. Once all the error bars are figured in, etc., the eclipse measurement was more likely to be wrong. So, it’s the type 2 prediction which gives us more reason to perk up our ears.

A true pedant might want to call the perihelion measurement a type 3 prediction, because it could be “predicted” (or at least “explained away”) by a dark matter hypothesis: an intra-Mercurial planet, which the people of the time called Vulcan. (Insert your own Star Trek joke here.)

Of course, now we have type 1 predictions — gravitational redshifts, to begin with — making the whole shebang more than a little academic.

Re: This Week’s Finds in Mathematical Physics (Week 246)

Blake Stacey wrote:

A true pedant might want to call the perihelion measurement a type 3 prediction, because it could be “predicted” (or at least “explained away”) by a dark matter hypothesis: an intra-Mercurial planet, which the people of the time called Vulcan.

Le Verrier predicted the existence of Neptune in the 1840s, due to anomalies in the motion of Uranus. It was found in 1846. Attempting to repeat his success, in 1859 he predicted the existence of a new planet to explain the precession of the orbit of Mercury. Between 1859 and 1878 there were some reported sightings of this planet, and it was dubbed ‘Vulcan’. But these sightings were never reliably confirmed. I’ve heard that by the time Einstein came along, fans of this hypothesis were reduced to positing a gaseous Vulcan to explain the precession of Mercury.

Shades of dark matter indeed!

Anyway, the true pedant could argue that almost every really interesting type 2 prediction is really a type 3 prediction, because someone, somewhere, has some nutty theory that already predicts this number. For example, there are certainly plenty of crackpot numerologists who can ‘explain’ the masses of elementary particles!

So, perhaps type 2 should be defined as:

2. The theory lets us calculate some quantity whose value we previously knew, but could not calculate using previous accepted theories.

But I don’t have the patience to fill all the other loopholes one can dream up, so please don’t point out more!

Re: This Week’s Finds in Mathematical Physics (Week 246)

We can imagine three things happening:

Yes! Certainly. I agree. I wrote essentially the same, necessarily in other words, a few comments above:

Technically I think it is quite right to say that string theory predicts gravity. […] we may tend to dislike stating this technically correct statement this way, because we may feel that it does not sufficiently amplify the point that this prediction is rather useless, for our practical purposes. #

So we all agree that “predicts/explains gravity” does not imply “get too excited”.

What I was just trying to point out is that the reverse is also false: “don’t be excited about it” does not imply that “string theory does not predict/explain gravity”, which was the statement I was responding to #.

For me, it is important that I understand what is true and not about the technical aspects of a given theory and know how to distinguish that from assessing what that implies for the relevance of the theory for our endeavor of understanding the universe.

I think it is a true fact that string theory predicts/explains gravity. You and Peter Woit keep emphasizing of how little use this is, for our practical needs (that’s what I meant by the “sociological” implication of this fact). And I agree with that!

Re: This Week’s Finds in Mathematical Physics (Week 246)

Having said what I said above, concerning how very non-exciting aspects of string theory are, I would want to add the following:

this is true from a phenomenological standpoint. Everybody who cares about observable physics but not about theoretical structures and the like, should ignore string theory.

On the other hand, personally, I am of that other kind: my main interest is maybe more the structural aspect of theoretical physics, than cranking out numbers and compare notes with the accelerator people.

From that point of view, I do find string theory very exciting indeed. It may be phenomenologically unviable, but I can hardly ignore it when I am interested in structural aspects of quantum theory, gauge theory and gravity.

I am often puzzled by conversations like

A: “AdS/CFT is considered to provide a non-perturbative definition of quantum gravity on asymptotically Ads5×S5\mathrm{Ads}_5 \times S^5 spaces.”

B: “Yawn, oh how very boring! That’s not the number of large dimensions we observe, nor the right sign of the cosmological constant. That’s so uninteresting. ”

Re: This Week’s Finds in Mathematical Physics (Week 246)

From that point of view, I do find string theory very exciting indeed. It may be phenomenologically unviable, but I can hardly ignore it when I am interested in structural aspects of quantum theory, gauge theory and gravity.

Urs, would you agree with the following two assertions?

1. GR is both background independent and local, in an appropriate sense. In contrast, no (known) formulation of string theory fulfills both desiderata: perturbative ST is not background independent, and AdS/CFT is not local.

2. Infinite conformal symmetry in the strict sense is not compatible with locality neither. To have local observables, i.e. correlators that depend on separation, you need an anomaly.

Being interested in structural aspects of quantum gravity, I find it very exciting to be able combine background independence and locality.

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A: “AdS/CFT is considered to provide a non-perturbative definition of quantum gravity on asymptotically AdS5×S5AdS_5 \times S^5 spaces.”

B: “Yawn, oh how very boring! That’s not the number of large dimensions we observe, nor the right sign of the cosmological constant. That’s so uninteresting.”

To understand the reaction, you need to imagine not just one person saying statement A, but hundreds of people writing papers about it — and seeking tenured jobs on the basis of these papers. An unproved conjecture can be fascinating but still become tiresome when enough people write about it. When jobs are at stake, the negative reactions can become more stronger.

This dramatic reaction to Maldacena’s exciting idea lasted for many more years — it goes on to this day. This naturally caused a strong counter-reaction from people who wondered why so many physicists (rather than mathematicians) should be getting jobs for working on supersymmetric quantum gravity in a universe with the wrong number of large dimensions and with a cosmological constant of the wrong sign.

Imagine, for example, that one year the second-best-cited paper in physical chemistry concerned a conjecture about a very beautiful theory in which carbon was a noble gas.

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To understand the reaction […]

Yes, I know: it is the “sociological component” of this which is disturbing. (Maybe that word is not the best one: I just mean it is a problem with us, not within the platonic world of ideas that string theory lives in, hope you see what I mean)

Just for myself, though, I will keep finding fact X, interesting indepent of the number of other people doing so.

And, as you know, for many of the facts that I find interesting, the problem is exactly the opposite as for the one we discussed here: there are annoyingly few other people appreciating them, rather than annoyingly many. :-)

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And, as you know, for many of the facts that I find interesting, the problem is exactly the opposite as for the one we discussed here: there are annoyingly few other people appreciating them, rather than annoyingly many. :-)

Amen, brother.

And then the two can go together. Few things are more demoralizing than an underpopular great idea (say, extensions of knot invariants to tangles, cospan extensions…) paired with a vastly more popular idea (say, Khovanov homology). The upshot is like the counterintuitive effect when two balloons are connected by a tube for air to flow: the smaller balloon contracts even further until none of the molecules left trying to fill it can get a job anywhere.

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Do I understand correctly that your concern here is that Khovanov homology is just a very specific example of the more general problem of extensions of knot invariants?

Partly yes and partly no. Khovanov-style homology theory is one form of categorification for knot invariants, and categorification is one (very important) part of extending knot invariants to tangles, but I wouldn’t say it’s just a special case. It lies in the intersection of the categorification program and the tangle theory program, and so goes beyond both in certain ways.

There are also “lower” parts of the tangle theory program that can lend insight into the Khovanov program. For instance, there are many tangle extensions of the bracket, and Kh(T) categorifies only one of them. And though it’s solved some old problems, I do agree with the old knot guard that cry out, “but where’s the topology?” The bracket extensions “on the ground” show all the possible shadows for categorification, and one of them may show what the topological content of the bracket is.

As it is, though, I know just a handful of people who have expressed interest in extending knot invariants to tangles, and only one person actually working in earnest on the problem. Everyone already knows that Khovanov homology is interesting now that he crossed that valley, and many people are busy climbing the hill. Trying to convince people to cross another valley is difficult at best.

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I don’t have the relevant books to hand, but I seem to remember an argument to the effect that scientists were more impressed by GR’s retrodiction of Mercury’s behaviour than they were by the bending of star light, and that they were right in this as Mercury tested GR more severely.

The argument was conducted in terms of number of parameters tested. For the sake of argument let’s simplify the situation so that we have a theory which when we ensure it lines up with an old theory in a limit fixes its constants. Then we may imagine that as we look at the expansion as a Taylor series about the classical solution there are a string of predicted coefficients. Now to the point, it may happen that a piece of retrodiction puts more of these coefficients to the test than does a prediction. In which case the retrodiction was a more severe test and has greater confirmatory power.

Of course, if you’re viewing all this from the outside and you see a new phenomenon - like light-bending - predicted and found, you’ll probably be more impressed than by an old phenomenon - like Mercury - explained. Without knowing the details of parameter-fixing this is reasonable.

Internally things are more subtle than I’ve allowed so far. You’d want to have an idea of how the selection of a particular theory from a family is done. Was it a ‘natural’ family, the chosen member of which is a simple choice by say agreement in the limit with an old theory. How much support do the principles guiding the choice of that family have. The fear being that the family has been engineered to look like a simple choice has been made to agree with the old theory.

A final note, those of you who have been reading my posts and/or cake talk on statistical learning theory will know that learning is not just about the number of parameters.

Re: This Week’s Finds in Mathematical Physics (Week 246)

John suggests 3 things that may increase our confidence in a scientific theory:

The theory lets us to calculate some quantity whose value we didn’t previously know. We measure the quantity and the theory turns out to be right.

The theory lets us calculate some quantity whose value we previously knew, but could not calculate using previous theories.

The theory lets us calculate some quantity whose value we previously knew, and could calculate using previous theories.

Item 1. (and only item 1.) is what I understand by the word ‘prediction’. If the quantity predicted and measured had been in some way determined earlier (but unknown to the proponents of the theory, for whatever reason), then you might say ‘retrodiction’ instead, but this is not philosophically signifcant.

I would call item 2. ‘explanation’, rather than ‘prediction’. Item 3. may be an explanation as well, if there is some superiority of the new theory over the old: initially, if it simpler (or background independent, or otherwise preferable, possibly controversially so); later on, if we come to believe (probably through separate evidence of type 1.) that the old theory is simply wrong and the new theory is (more) correct.

Well, this is how I understand the words ‘prediction’ and ‘explanation’, but I won’t fight for them. Now that John has made this list, it’s probably better just to use the numbers 1,2,3., since they get at the heart of the matter, rather than arguing over the meaning of common words.

And here is one point at the heart of the matter: No matter how many successes a theory has of type 3., and even of type 2., in science we require confirmation of type 1. as well. If necessary, we force events of type 1. to occur; that is, even if we have already accounted for all observations, we perform experiments to create new observable phenomena. Without experimentation (or some other source of continually new observations), we do not have falsifiability (to use Popper’s term), at least not in practice. Then we risk degeneration into ‘Greek science’ (if I may be so boldly insulting to Aristoteles and company), flying into the clouds of theory without the ground of observation. (And this problem may occur through no fault of the theory or its proponents, if we simply lack the technological ability to perform experiments! Therein lies the tragedy of this endeavour.)

Re: This Week’s Finds in Mathematical Physics (Week 246)

Well, this is how I understand the words ‘prediction’ and ‘explanation’, but I won’t fight for them. Now that John has made this list, it’s probably better just to use the numbers 1,2,3., since they get at the heart of the matter, rather than arguing over the meaning of common words.

Agreed! What’s more, by speaking of 1-dictions, 2-dictions and 3-dictions, we can generalize to nn-dictions, where the “worth” of the diction falls with increasing nn. This gives a numerical gloss to the qualitative graph shown on page 5 of this Alan Sokal paper.

Re: This Week’s Finds in Mathematical Physics (Week 246)

David Corfield wrote:

Many historians of physics reckoned that general relativity’s accounting for the anomalous precession of Mercury’s perihelion was hugely influential in getting physicists to sign up to it. Now what kind of ‘-diction’ was involved there? The anomaly had been known for several years.

You already said yourself what it was: an explanation. That’s valuable, but that’s all; it’s intellectually dishonest to try to turn this into a prediction. (And GR had its own important prediction at the time: the bending of light by massive objects).

Philosophers of science usually call it a retrodiction.

Do they? This doesn’t fit my own understanding of the word’s meaning (which is well represented by the definition that I quoted from Wikipedia). Perhaps people do not use the word consistently?

Did it matter whether the data had been used in the construction of the theory, etc. Indeed, Einstein had rejected an earlier theory because it was incompatible with this data.

In my opinion, this is exactly what is important. If Einstein hadn’t known about Mercury, then this should have counted as a prediction (or retrodiction, but as I said the distinction is vague and unimportant) for him, increasing his own confidence in the theory. As it was, Einstein was still rightly confident, as only his theory provided an explanation, but not as confident as a confirmed prediction should make him. In any case, astronomers at large would count this only as an explanation, not a prediction. But if they too had not known about Mercury’s anomaly (say, if they were just beginning to get such precise measurements), then they too would count it as a prediction, and it would have been a more stunning success.

String theorists may argue that string theory explains gravity. (I don’t think so, since the explanation is more complicated and less clear than Einstein’s theory, which is to be explained. But at least it is consistent with quantum physics, so I can certainly accept a difference of opinion here.) But unless they’re using words differently from the way that I understand them, it certainly does not predict (or retrodict, whatever) gravity.

Science is more than merely a search for explanations of known facts. It relies also on tests against new facts. This is where prediction comes in.

Re: This Week’s Finds in Mathematical Physics (Week 246)

String theorists may argue that string theory explains gravity. (I don’t think so, since the explanation is more complicated and less clear than Einstein’s theory, which is to be explained.

At the classical level, which is what you seem to have in mind, Einstein says: extremize the Einstein-Hilbert action.

String theory would offer an explanation for why this is the action functional to be extremized: because this is what makes spacetime a target for conformal 2-dimensional theories.

That, in turn, would happen to xxxdict or explain the presence of a perturbative quantization of the Einstein-Hilbert action.

It might be useful to look other examples for this kind of explanation of one theory by another.

Alain Connes has another theory which offers an explanation for why Einstein found himself extremizing the Einstein-Hilbert functional, instead of some other functional: he postulates that the action functional in question should be a functional of target space together with a certain Dirac operator on that, satisfying a natural compatibility condition. He calles this principle the “spectral action principle”.

And it predicts/explains gravity: feed in a Dirac operator and out comes the Einstein-Hilbert action functional, together with couplings to other fields.

What Connes’s principle so far does not predict is the existence or nature of a quantum version of this.

Anyway, I very much agree with what you say about the usage of “predicts”, “explains”, etc. as long as the everyday understanding of these terms is concerned, and their implications for the gullibility of those who have to distribute the money, but for technical reasons I doubt that you would want to consistently search for alternatives for “predict” when you want to refer to the implications some physics theory has.

Well, I’d be content with consistently using any other reasonable term to express the technical idea “follows from the axioms”. “Explains” would be fine with me.

Re: This Week’s Finds in Mathematical Physics (Week 246)

My choice, in chapter 6 of my book, was “accounts for”.

Okay, that’s another good possibility.

If all of physics had been made very precise, a physical theory would be a collection of definitions and axioms, and the only reasonable term for what the theory does to statements it explains, predicts or accounts for would be: “implies”.

As Toby points out, many physics theories are much more vague than our standard mathematical axiom systems. But, on the other hand, hypotheses and implications of these certainly have their place also outside of rigorous mathematics, where they may be subject to a certain degree of doubt, but still useful and relevant.

In fact, let me say: the assumption of a surface weighted by its proper volume to have a consistent quantization implies that its target space is a solution to Einstein’s equations (yes, in 26 dimensions and with a bunch of extra fields coupled to it).

Re: This Week’s Finds in Mathematical Physics (Week 246)

I’d be content with consistently using any other reasonable term to express the technical idea “follows from the axioms”. “Explains” would be fine with me.

Perhaps I’m just not properly aware of how words like ‘prediction’ are technically used in science (or rather, the philosophy thereof). But I certainly would prefer ‘explains’ to ‘predicts’ here.

Actually, I don’t intend anything so formal or mathematical by ‘explain’, so a theory can explain a fact without being precise enough that following from axioms is an applicable concept. On the other hand, a complicated theory doesn’t really ‘explain’ a simple fact, even when the fact clearly does follow in as strict a sense as you may desire, so there is something akin to algorithmic complexity (as you suggested) involved as well.

In any case, these are sociological or philosophical terms rather than strictly scientific ones.

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On the other hand, a complicated theory doesn’t really ‘explain’ a simple fact

I take it you are arguing that string theory does not even “explain” gravity (let alone predict it) due to it being more complicated than Einstein gravity. Is that what you have in mind?

This is curiously opposite to how I perceive the situation. String theory may be all wrong and everything, but it sure derives gravity (and a little more, maybe a little too much ;-) from a very simple premise:

The action of the plain relativistic particle (in Nambu-Goto form) is the proper worldvolume of its worldline. Quantizing this gives the Klein-Gordon equation (the relativistic Schrödinger equation). Second-quantizing this gives free field theory.

The single premise of perturbative string theory is to replace the 1-dimensional worldline of the Klein-Gordon particle by a 2-dimensional surface, while naturally keeping the action to be given by the proper volume.

That’s it. Nothing more. Nothing more natural than this generalization.

Quantize this, and you run into a very rich structure, including gravity.

Re: This Week’s Finds in Mathematical Physics (Week 246)

The single premise of perturbative string theory is to replace the 1-dimensional worldline of the Klein-Gordon particle by a 2-dimensional surface, while naturally keeping the action to be given by the proper volume.

That’s it. Nothing more. Nothing more natural than this generalization.

Quantize this, and you run into a very rich structure, including gravity.

“Nothing more”?

If you just “quantize this” you have a problem on your hands with a tricky symmetry to handle. When you work hard and figure out how to handle it you have a 26 dimensional theory with a tachyon that has no stable vacuum state.

Now if you figure out how to instead construct a superstring (which requires more new ideas than “nothing”), you’ll have a ten-dimensional theory. Sorry 10d GR is not the gravity we know and love that Einstein figured how for us.

Getting successfully from 10d down to 4d is not exactly “nothing”.

Again, string theorists who claim that superstring theory predicts gravity should make clear when they say this that they are talking about 10d gravity.

Re: This Week’s Finds in Mathematical Physics (Week 246)

I take it you are arguing that string theory does not even “explain” gravity (let alone predict it) due to it being more complicated than Einstein gravity. Is that what you have in mind?

Yes, this is what I have in mind. But a couple of caveats:

First, my opinion that string theory doesn’t really explain gravity is not a strong opinion. (Remember, it was just a parenthetical comment orginally, within a philosophical discussion of the difference between explanation and prediction.) I think that Peter Woit makes good points (such as those in the comment immediately above this one), but really you should discuss it with him, since I know much less about the matter than you two do.

Second, to return to the philosophical discussion, I think that it’s quite possible for a theory to predict something that it does not explain! Well, maybe not at the same time, but the relevant chronologies are different. To take a somewhat artificial example, consider this word problem from the textbook for the Algebra course that I’m teaching these days:

Find the annual interest rate on a savings account that earns $110 in 1 year on a principal of $1000.

According to the theory embodied in the techniques taught in the textbook, the interest rate is 11%. Of course, the interest rate has already been set, but since we don’t know what it is before we work the problem, this counts a bona fide (and scientific) retrodiction, which is philosophically just as good as a prediction. But is it really fair to say that these techniques (even together with the data in the problem) explain the interest rate? They do explain why 11% is the correct answer to the word problem, but if this problem describes a situation in the real world, then we should look to banking policies and economic forces, rather anything involving the specific number 110, to explain the interest rate. In other words, we eventually want a deeper understanding for the explanation. Nevertheless, to test whether algebra works (or whether the data are correct), the prediction (if confirmed) stands as a valid success.

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I think that it’s quite possible for a theory to predict something that it does not explain!

Of course. The theory ‘It always rains on a Thursday’ predicts that it will rain next Thursday, but it doesn’t explain it.

The reason I opted for ‘accounts for’ rather than ‘explains’ is that there’s is an enormous literature on scientific explanation, e.g., Wesley Salmon’s Four Decades of Scientific Explanation.

Here’s a good place to start you off. Which way does a helium filled balloon tilt when it is held by a string in the hand of a passenger during take off?

You’ll all no doubt correctly say ‘forwards’. Now why?

1) Because of the pressure differential created while the plane accelerates.

2) As being accelerated is equivalent to experiencing a gravitational force, the horizontal situation in the plane is equivalent to the vertical situation while lying on your back in a room. In the latter case, the balloon rises.

Re: This Week’s Finds in Mathematical Physics (Week 246)

David asked at last:

Are these both explanations?

Well … being blissfully ignorant of the voluminous literature of explanation, I may end up being quite naive, but as long as you don’t mind that …

I’d say that (1) is an explanation to somebody who already understands about air pressure and so forth, while (2) is an explanation to somebody who already understands about the principle of equivalence (and also understands, somehow or another, that helium balloons normally rise). This is because the explanation is supposed to make the description simpler, so we need to already understand the background material.

It is (background material) + (application to explain this situation) that is simpler than (background material) + (merely observed or claimed phenomenon); if we had to explain all of the background material as well, that would end up being more complicated than (observed or claimed phenomenon alone). On the other hand, (theoretical material) + Σi (application of this theory to phenomenon i) is liable to be simpler than Σi (independent observed or claimed phenomenon i), so you doubtless want to explain the general theory (the existence of air pressure, the principle of equivalence, etc) eventually, even to somebody that doesn’t already know that stuff.

There’s also an interesting relationship between (1) and (2). Assuming that somebody already understands the air-pressure explanation for why helium balloons rise in ordinary life, we can now apply the principle of equivalence in a very direct way to that explanation to produce the air-pressure explanation for the motion of helium balloons in accelerating cabins. That is, (principle of equivalence) ° (air-pressure explanation why balloons rise) = (air-pressure explanation why balloons move forward in accelerating cabins), where ° is function application in some sense.

(I’ve thought about this idea —applying a general principle to a concrete explanation to produce another concrete explanation— before in the context of proof theory. It works pretty well in the proof theory of constructive mathematics; this is the basis for the idea that a constructive proof yields a computational algorithm, an idea that has been implemented, for example, to turn proofs in Coq into programs in Haskell. It should work here too, except that the concepts are all less precise.)

Re: This Week’s Finds in Mathematical Physics (Week 246)

Well, I’ve never heard of the idea you raise in the last two paragraphs. It’s an intriguing thought.

As for what they do talk about, the two major currents in explanation theory are

(1) Subsumption of facts under general laws.

(2) Derivation of facts in terms of causal mechanisms.

My example was given with these in mind. As for (1), while initially the hope was that only the syntactic form of a statement is relevant to its lawlikeness, this proved not to work. There’s something more to a law than its being a true general statement. (I touch on this in my natural kinds paper.)

Some responded to this by trying to cash out explanation in terms of the derivation of disparate facts under a unified theory. There’s a question here of whether one has gone beyond what you might call ‘descriptive economy’ of law to the detection of (real) ‘natural kinds’.

Those who follow (2) also want to go beyond the positivists by talking about causal mechanisms.

But what you write points to a third strand - the pragmatic dimension. That an explanation is an answer to a why question. That a why question presupposes a contrast, comparing a description of what happened with what did not happen. And that the giving of a satisfactory explanation depends on the state of knowledge of the receiver.

Re: This Week’s Finds in Mathematical Physics (Week 246)

Pet peeve of mine, but I wish people would not use the word “quantization” to describe a process which results in a classical (field) theory. The words “relativistic Schrodinger equation” have confused generations of graduate students as well…

Re: This Week’s Finds in Mathematical Physics (Week 246)

Not sure why you say so, since you seem to be exactly referring to the issue first/second quantization.

We are quantizing the Nambu-Goto and/or Polyakov action (either in their 1-dimensional incarnation (relativistic point particle) or their 2-dimensional incarnation (bosonic string)) to find a quantum theory on a 1- or 2-dimensional parameter space.

The “relativistic Schrödinger equation” here is the 1-dimensional counterpart of Ln|ψ⟩=0L_n|\psi\rangle = 0 for the string, and that’s quantization.

Of course, we may alternatively regard ∂2|ψ⟩=0\partial^2 |\psi\rangle = 0 as a classical equation and quantize again. That leads to a field theory on the former target space.

Analogously, we may, alternatively, regard Ln|ψ⟩=0L_n |\psi\rangle = 0 as a classical equation and quantize again – that leads to string field theory on the former target space.

Re: This Week’s Finds in Mathematical Physics (Week 246)

Well, the Klein Gordon equation, even if you write it as the Klein Gordon operator acting on a ket, is still an equation for a classical field, not for a wave function. For start that field is real, and the square of it is not probability of anything. Planck constant is zero throughout.

The only similarity is that both Klein Gordon and Schrodinger equations are differential equations, but physically they have nothing at all in common. Similarly first quantization of the string yields classical string theory, there is no place for h-bar anywhere.

Physically, there is really only one “quantization”, where you introduce wave-functions and probabilities, all this talk about first or second quantization and what not is just a good way of getting confused.

Re: This Week’s Finds in Mathematical Physics (Week 246)

Well, the Klein Gordon equation, even if you write it as the Klein Gordon operator acting on a ket, is still an equation for a classical field, not for a wave function.

The space of normalizable real solutions of the Klein–Gordon equation can be made into a complex Hilbert space in an essentially unique Poincaré-invariant way.

In fact, this complex Hilbert space is a irreducible unitary representation of the Poincaré group!

Such representations were classified by Wigner, who showed they correspond to various sorts of particles. The space of real solutions of the Klein–Gordon equation is the Hilbert space of a massive spin-0 particle.

One can then form the Fock space on this Hilbert space, which is the Hilbert space for arbitrary collections of identical massive spin-0 particles. Equivalently, it’s the Hilbert space for a massive spin-0 free quantum field.

It may seem odd that the space of real solutions of some equation is naturally a complex Hilbert space. But, there are lots of ways to see this. One is to show that the space of normalizable real solutions of the Klein–Gordon equation is isomorphic to
L2(ℝ3)L^2(\mathbb{R}^3) — the space of complex wavefunctions on space. Under this isomorphism, time evolution is given by the Hamiltonian that’s the natural relativistic generalization of the Hamiltonian for Schrödinger’s equation:

H=−∇2+m2 H = \sqrt{-\nabla^2 + m^2}

in units where ℏ=c=1\hbar = c = 1.

This stuff is ‘well-known’… but often neglected in textbooks! I explained it in more detail here.

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John, thanks! I have not thought about it this way. The way I present this when talking about first “quantization” (and I do use the scare quotes) is through the heat kernel, or Schwinger proper time method, of solving linear differential equations.

In this method your differential equation is replaced by some heat (diffusion) equation for the kernel, or if you are not fussy about factors of (i h-bar) by Schrodinger equation. The time there is some auxiliary parameter, which you can visualize if you want as parameter along the worldline of some particle (though personally I think it should inherently be thought of more as Euclidean time).

So, after that we have all the machinery of Hilbert spaces, operators, Hamiltonians and path integrals and all that good functional analysis stuff. Since we physicists only see that machinery in quantum mechanics class we conclude immediately (and erroneously) we quantized something. I just find that this language leads to confusions regarding physical interpretation. Maybe it is better not use “quantize” or “wavefunction” etc. when the objects we discuss are somewhat similar mathematically but have totally different interpretation.

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I’d conclude I quantized something when I did go through the quantization procedure.

We find ∂2ψ=0\partial^2\psi = 0 from starting with the action of the free particle and either doing the path integral, or in fact the canonical quantization.

similarly first quantization of the string yields classical string theory

I guess you’d get a negative reaction to this statement if you told that to somebody working on the conformal field theory on the string’s worldsheet.

Of course I know (well, at least I think so, you will please correct me if not) what you have in mind: the effective theory on the target space of the string (its string field theory) is still classical if we only have the worldsheet theory quantized.

But that’s exactly what the step from “first” to “second” quantization is all about.

On the other hand, I’d perfectly agree that the terminology “second quantization” is not the best one.

Remarkably, this is all at the very heart of the entire program of perturbative string theory, which says:

define a quantum theory on target space by specifying its perturbative expansion as a sum over diagrams which are themselves weighted by numbers obtained from a(nother) (nonperturbatively defined) quantum theory in 2-dimensions.

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Yeah, OK, this is all about semantics anyhow. I am fine with referring to the familiar mechanical procedure as “quantization”, whether or not it involves actual quantizing, as long as one is careful about the precise interpretation of what is going on. Perhaps just a warning that “quantization” does not always entail quantum mechanics (you know, probabilities, interference, Bell inequalities) is sufficient.

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Oh, one more thing. When arriving at a worldline through the heat kernel method, the Hamiltonian would be simply the Klein-Gordon operator. In fact one gets a gauge fixed version of the generally covariant worldline action, with e (the einbein) already set to 1. The Hamiltonian you mentioned is obtained from the same generally covariant action upon different gauge choice, where the worldline time is chosen as the spacetime time. In other words the two choices are related by worldline reparametrization.

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In your comments on the two books about the crisis in high energy theory, you consider the situation as if the community of theorists were independent of outside world. I think it may be useful to take into account that the crisis may be a part of a general trend. In this trend the average mark obtained by scientists from the society dropped (20 years ago it was easier to boast that you are a scientist).
It happend for reasons independent of science performance. So, a substantial fraction of the theorists have been struggling to maintain the high score of science in the public eye. And it turned out that the ground was too hard at that time: if we say of successfull computations for high energy particles, not much have changed the last 20 years. So, the folks were desperate to invent something flashy. String theory was flashy in 1983 and it is flashy now, while the success of the ‘modest’ standard model combined with perturbation theory remains to be the only undeniable achievement of particle theory. One can draw a moral from this story: science is not an easy bread, and one taking it as ones occupation should probably not count on becoming a succesful middle class person.

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Hi John,

a very interesting post. Your analogy with Vegas is in a certain regard quite ironic. I was at the String Pheno at KITP last year, and of course there was a lot discussion about the landscape etc, that our world might only be one in a huge multiverse. In one of the sessions the comment was made (sorry, can’t recall who) if someone would find the point in the landscape that reproduces the SM, then we could just work with it and forget the rest, 20 years ago nobody would have been bothered by ‘the rest’ if we only had a theory that worked.

Being pragmatic myself, I am kind of sympathetic to this opinion. If we had a working model with all the promised blessings of string theory, then lets just use it, no?

The problem is, to come back to the Vegas analogy, that I can imagine generations of grad students playing the landscape lottery trying to get lucky and to find one that is ‘just right’! And that’s definitly not where I want theoretical physics to go.

Another thing that I’d like to add is that the argument with the valley climbers and mountain seers (or swhatever) is a nice one, but one shouldn’t forget that as everything in life it is a matter of balance. Besides the seers we still need the craftsmen, and the valley crossers need the mountain climbers - especially in a global community that is as tightly connected as ours. It is totally silly, and equally irresponsible, to keep on doing science in the 21st century without taking into account how much the world, and our community with it, has changed.

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Hi Christine,

There is plenty of room for all of them. If there’s only one point in which I agree with Lee 110% then it’s that theoretical physicists aren’t expensive. Compare that to the money that goes into experimental physics, or worse - military applications. Sometimes I want to scream at the funding agencies to just add .1% to the total and give it to me, gee, how many people could be hired with that?! It’s so totally stupid that positions are so rare because it’s totally unnecessary. The problem is (I think) that most positions still are at universities and tied to teaching, of which there is limited demand. What we really need are more pure-research positions. But these are currently mostly provided by private institutions. Best,

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Hi Bee,

Yes, of course. In my comment, “suitable” means exactly what you say – “pure-research positions”. And when I ask whether there would be “room for them all”, it is more like an ironic question, given that people want to put money (even given that it’s far from being an “astronomically” high investment) elsewhere. Indeed, it wouldn’t cost too much to change the situation considerably.

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Hi, I remember John Baez from old arguments on the sci.physics newsgroups when he defended physics against all sorts of crackpot theorists, some quite nasty as I recall. His was, and I’m sure still is, a very welcome voice of both reason, knowledge and authority. (sic) In that order! So it’s nice to see you have a blog, John and I’m glad I found it.

It’s a long time since I posted anything on physics (I’m not a physicist), but now I’m reading Lee Smolin’s book and can’t help myself. The first part is great, one of the clearest explanations of certain fundamental issues I’ve ever read.But then when he starts talking about alternatives to the string theories he’s so neatly skewered, he starts sounding like part of the problem he’s trying to fix.

It seems to me that the sort of fundamental issues Smolin is addressing have definitely reached a crisis point – and it is certainly a crisis for physics, because it seems to me that physicists are no longer really prepared anymore to address such issues. It’s as though the current faith in math as the cure-all for everything scientific is starting to remind us of the old faith in, say, perfect circles as the ideal path for planetary orbits to take.

For example, take the question Smolin raises concerning certain basic constants, such as the particle masses. At a certain point I kept hearing myself say, “evolution, evolution, evolution” and then lo and behold HE starts talking evolution – but in a really weird way. Suddenly instead of simply evolution itself, which strikes me as being by far the most likely answer, he’s talking about some “multiverse” that as far as I can see is every bit as dubious and unfalsifiable as any string theory. If I were a physicist I suppose it would look natural to me but I’m not so it doesn’t, it looks just very weird. As weird as all those strings.

But if you want an explanation of all those masses, then why not consider just plain old evolution without all the fancy stuff about multiverses? Does EVERYTHING in physics have to be part of some really neat, “beautiful” scheme where it all fits into neat little channels like Kepler’s orbits based on the Platonic solids?

I think those masses are the way they are because the universe evolved (THE universe, not some wacky multiverse) – and just as in biological evolution certain very unlikely things just happened and then for one reason or another propagated themselves. I read recently that the only reason animals have mouths is because some ONE organism a few billion years ago experienced a mutation that produced something that eventually became a mouth. And if that organism had not experienced that particular mutation and its progeny had not survived, then we would not have mouths today, but something else. I think those mysterious constants are most likely derived from exactly that sort of random event. I have a feeling that’s what Smolin thinks also, but for some reason he felt he needed to bring other stuff into it and the most interesting and significant and in fact revolutionary aspect got lost in the mathematical fog.

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I think those masses are the way they are because the universe evolved (THE universe, not some wacky multiverse) – and just as in biological evolution certain very unlikely things just happened and then for one reason or another propagated themselves.

We are obliged to carefully analyze the collections of constants that go into the standard model of particle physics in order not to miss a hidden fundamental structure it may have, which may reduce its algorithmic complexity and point us to a theory going beyond it.

We may even hope that this is the case.

But we have no good reason to be surprised if none such structure exists.

Not any more than Kepler had good reason to be surprised to find out that the constants of the solar system do not follow the laws of Platonic solids.

In general, in physics: expect your laws to be beautiful but the data to be messy.

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The idea of evolution is a great distillation of the concept which Urs mentions: beautiful laws and messy data. As Dawkins put it, life develops through the non-random survival of randomly varying replicators. The complexity of a living thing is the reflection of the environments in which its ancestors lived, the messiness of the past environments borne out in the intricate genome.

So, if we consider the Universe as an evolved thing, what are the “replicators” and what forces affect their “survival”? What is the analog of genetic information, and how is it expressed? Are there counterparts to the “selfish gene” model, group selection, viruses?

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Privately, over a beer, I would enjoy speculating about ideas like the universe replicating itself through black holes, or some other far-fetched thing like that, but in the present context I feel a little reluctant to do so, with already the plain facts leading to the kind of unfruitful discussion we see over the blogosphere.

In as far as I supported the idea of the data of the standard model as a result of an “evolution process” I had in mind just ordinary “time evolution”, expressing the mere fact that things do follow their laws of motion over time.

While Darwinian evolution, like any mechanism with strong feedback channels, is a nice example for how a simple law can lead to highly involved output, I would not think that it is a good analogy for the topic we are discussing.

The shape of the Alps, say, is certainly a result of a natural process, and of “time evolution”, but certainly not of a feedback process involving anything like mutation and selection. Still, nobody would want any theory to predict, postdict or retrodict the shape of the Matterhorn, characteristic as it may be.

And, please note well: I am not saying that it is clear that the masses and couplings of the standard model are not more like the shape of a crystal than that of a mountain. I am just saying that, at the moment, we have no good reason to expect to see more crystals than rocks.

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I agree, the “feedback” aspect of Darwinian evolution, aka “natural selection” doesn’t necessarily apply to cosmic evolution. But the latter may well be analogous to a very different aspect of evolutionary thought currently being applied in population genetics: the study of neutral markers.

Mitochondrial DNA, for example, is important not only because it doesn’t recombine, but because it is also presumed not to participate in the selection process, making it a much better index of population history/lineage.

What apparently determines ones mtDNA is simply random mutations and the inheritance of same unchanged over many generations. What apparenty determines the mtDNA statistics of a population are things like founder effects produced by population bottlenecks produced by very specific, often catastrophic, events. There would seem to be no equivalent in cosmic evolution to mutations, but there certainly would be the possibility for founder effects/bottlenecks produced by catastrophic events during the very early stages of the Big Bang.

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Having made some anti-string comments above, I want to balance those here. We have no right to expect any simpler explanation of the world than a random point in a space of possibilities. It will be extremely cool if string theory (only!) gives us a framework for understanding the evolution of the universe; it would be interesting even if we couldn’t test it, but absolutely fantastic if we could and it was successful. People should keep thinking about the Landscape and the other ideas of string theory; they just shouldn’t say that anything has been solved. (And of course, they should think about other things too, but I believe that all of us here agree on that point.)

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Hi Victor,

I’m not John, but have a comment about your sentence

I think those masses are the way they are because the universe evolved (THE universe, not some wacky multiverse) – and just as in biological evolution certain very unlikely things just happened and then for one reason or another propagated themselves.

What exactly do you mean with ‘universe’ and are you sure you use the word in the same meaning as Lee does? See, we know that in our observable part of the universe the parameters of the standard model can’t have varied very much. So where did the ‘unlikely things’ happen and where did they ‘propagate themselves’?

Best,

B.

PS: I have a feeling that’s what Smolin thinks also, but for some reason he felt he needed to bring other stuff into it and the most interesting and significant and in fact revolutionary aspect got lost in the mathematical fog.

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What exactly do you mean with ‘universe’ and are you sure you use the word in the same meaning as Lee does?

No, I’m not. And you’re right I should read his book about that, in fact I’m really excited to learn of its existence and will definitely check it out.

See, we know that in our observable part of the universe the parameters of the standard model can’t have varied very much. So where did the ‘unlikely things’ happen and where did they ‘propagate themselves’?

I was referring to the Big Bang and what happened right after that. I guess the picture I have in my mind is of some set of contingencies very early on that could have determined those parameters in ways we have no hope of reconstructing, at this late date. And once they were set, then they could have just remained fixed as the particles (or their precursors)exploded out beyond the point that any similar types of contingency could affect them. Does this resemble Smolin’s (or anyone else’s) idea? I hope so, because it seems very logical to me.

What’s been on my mind is a question stemming from the “Out of Africa” theory of human evolution. I.e., if we are all descended from the same “Ur” homo sapien, or small group of homo sapiens, then what is the source of all our genotypic and phenotypic differences? In very crude terms, why do Chinese people look Chinese and Europeans look European, etc.?

In his book “The Real Eve,” Stephen Oppenheimer offers an amazing answer in the form of a contingent event that we actually do happen to know a great deal about: the explosion of Mt. Toba, circa 72,000 years ago. This event, or something like it, happening in the early stages of the Out of Africa migration, could have led to a whole set of different population bottlenecks, and consequent “founder effects” that might possibly have been the source of such differences.

It’s an interesting idea for physicists to contemplate because there may be a parallel between “Out of Africa” and the Big Bang, from an evolutionary standpoint.

So, soon after the Big Bang, maybe something drastic happened, like Mt. Toba, that could have led to something like a particle or protoparticle “bottleneck,” with subsequent “founder effects” producing the particle zoo we contemplate today.

Geneticists can actually use the DNA of currently existing humans to make inferences about such bottlenecks.
So I guess the question here is whether physicists have some analogous historical tool(s) that would enable them to look back and make inferences about what happened just after the Big Bang. And I guess they do, because they have in fact made such inferences. But as with genetics, it may only be possible to take the research so far and no farther.

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Victor Grauer wrote:

Waddya think, John?

I think I don’t want to talk about this. But, I’ll say one thing.

Suddenly instead of simply evolution itself, which strikes me as being by far the most likely answer, he’s talking about some “multiverse” that as far as I can see is every bit as dubious and unfalsifiable as any string theory. If I were a physicist I suppose it would look natural to me but I’m not so it doesn’t, it looks just very weird. As weird as all those strings.

Whether a theory ‘looks weird’ is not the main point. What matters more is that it delivers testable predictions, which turn out to be right.

I think string theory doesn’t yet deliver testable predictions. More precisely, I see no great chance of string theory delivering an event of type 1 or type 2 anytime soon. The best it can do is give us type 3 events. Others may disagree, but to understand Smolin’s book you have to know that he agrees with this.

On the other hand, Smolin claims that his Darwinian cosmology has already delivered an event of type 2! He explains this here:

To summarize, he claims that his theory gives correct estimates on the masses of the proton, neutron, electron and neutrino, together with the weak, strong and electromagnetic coupling constants. String theory does nothing like this. So, from his viewpoint, he’s ahead of string theory.

Others will disagree heatedly, of course.

I don’t feel like getting into this argument again, so I’ll quit here. I just want you to see where Smolin is coming from.

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OK, I had a quick look at Smolin’s paper and found it very intriguing – though most of it was way over my head, natch. Actually the part about black holes giving rise to new “universes” interested me VERY much, because it resonates with an old crackpot theory of my own – but I don’t want to get into any of that quite yet. :-)

The main problem I have with his version of cosmological evolution is that it seems to still, despite himself, depend too much on the overall bias of the anthropic principle. In other words, it seems to me that both the anthropics and Smolin may be looking through the wrong end of the telescope.

Darwinian selection, i.e., “fitness,” is NOT about fitting something to the way things are at present, but about how things come to fit the environment in which they existed at the time the adaptation occured. The anthropic principle, it seems to me, is an example of what could be called the “destiny” fallacy, the notion that all things evolved according to the rules of some preordained system. So when Smolin looks at cosmic evolution from the viewpoint of how many black holes it would be likely to have produced and whether or not that jibes with the number likely to now exist, that too strikes me as the wrong way to think about evolution. In fact, it seems as though the whole notion of all these many (infinite?) universes is forced by the need to avoid the notion of predestination that’s already built in to the anthropic principle. To get around the very tendentious idea that the whole purpose of evolution was to produce US, it’s necessary to hypothesize “billions and billions” of universese which evolved differently. That, to me, is a measure of the inherent weakness of the anthropic principle – including Smolin’s attempt to get around it.

NOT that I’m against extrapolating from present knowledge to a theory about what might have happened in the past, that seems perfectly legitimate. But using our present situation as a guide to the meaning of evolution itself, THAT strikes me as a potentially serious fallacy.

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I don’t think anybody will recognize the right theory very quickly since good math and predictions apparently aren’t enough by themselves. There would probably have to be very precise predictions to get recognized quickly and there’s a very good chance that precise predictions would be computationally difficult even for the right theory. That the Standard Model has good tree level calculations to me means that string theory should have a GUT and all universes in a multiverse should be Standard Model ones. String theory or something else would be handling the corrections to tree level “above” the GUT. A Gross-like emergent spacetime also sounds perhaps like something that should be down in the GUT of a good string theory. Maybe supersymmetry has not been seen cause there’s none in the GUT and maybe if the GUT has its own spacetime then maybe the problems of the 26-dim bosonic string aren’t problems.

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What was it Thoreau said? Something like, “There are a thousand hacking at the branches of evil to one who is striking at the root.” Doesn’t everyone agree with Smolin that the root of this problem goes back to the beginning of science, to the mystery of how nature is both discrete and continuous?

Thousands are hacking at the branches of the unification problem, while precious few are taking aim at the root. It seems to me that one looking to the root is Abraham Fraenkel and company (see Foundations of Set Theory, 1973). They write:
Bridging the gap between the domains of discreteness and of continuity, or between arithmetic and geometry, is a central, presumably even the central problem of the foundations of mathematics.

If it’s the central problem of mathematics, then no wonder it’s the central problem of physics. First, numbers were thought to be “all,” but then the surds came along. Then, as Fredkin points out, matter was thought to be continuous, before discrete atoms were discovered. Electricity was thought to be a continous fluid, until discrete electrons were found. Light was thought to be continuous waves, but then discrete photons popped out. Finally, angular momentum had to be quantized in units of quantum spin.

Now the search is on for discrete gravitons, but if gravity is a consequence of spacetime, and quantum gravity means quantum spacetime, then shouldn’t somebody be looking at number theory first, where the whole continuous/discrete thing starts?

Maybe, the Cantor-Dedekind postulate that the geometric linear continuum corresponds to the arithmetized real continuum, is incorrect. Maybe, the way the physical continuum is modeled with abstract discrete entities is incorrect. If so, it follows that what physicists need is a better number theory, not an ad hoc invention like vibrating strings. Volovich’s has some compelling ideas along this line (see: V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, p-Adic Analysis and Mathematical Physics (World Scientific, 1994).

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OK, great, thank you, Doug, because this is another very important aspect of the issue raised by Smolin that I wanted to address. But probably not in the manner you were anticipating – since I see no hope whatsoever for reconciling continuity and (radical) discontinuity. In fact I am going to say something that I really hope doesn’t offend everyone here, because that is NOT my intention: the fundamental issues pointed to by you – and by Smolin as well – cannot be addressed by physics and, in all likelihood, physicists are NOT prepared to deal with such issues. Not because physicists are inherently incapable of that, but because it is something that is simply not part of either their training or their experience.

What is really at stake here is NOT the question of what space and time really really are in the most fundamental sense, but the even more fundamental question of how it is possible to represent them. In other words, we are dealing with not only epistemology but semiotics, the “science” of representation.

The only physicist I know of who really deeply understood this was Bohr, whose notion of “complementarity” is, at base, not a part of physics or even science, but is an essentially semiotic principle through and through.

That’s all I have time for now, but if anyone wants to pursue this line of thought, let me know and I’ll continue.

The appealing thing for string theory for me is the way the story is told for the very very general audience:

1. Replace point particles by more general geometric objects,

2. These geometric objects are strings, open or closed and later even more complicated gadgets.

3. To make the theory works without troubles, a new form of symmetry - supersymmetry is required.

4. For the theory to work requires additional six dimensions. Our universe has 10 (or even 11) dimensions.

5. Various remarkable symmetries and connections between different string theories are gradually discovered.

String theorists has the feeling of a big puzzle being slowly solved and the partial unfolded picture being beautiful: Their excitement and devotion is moving. It appears that string theorists agree the puzzle is bigger than expected and the progress is slower than expected.

Those who do not think string theory will prevail may still think that some pieces will be useful for physics or at least for mathematics. For example, particles which are not point particles; extra dimensions.

Two questions I am puzzled about. The first is: Can strings be fractals? In all the blogs/popular accounts the pictures describe strings as smooth and nice. Is it part of the theory or only part of the pictures?

The second is: Is there such a nice short description that can appeal to a very very large audience for loop quantum gravity. OK, maybe it is a mistake to worry at all about the very general audience and not the very savvy experts, maybe the popular accounts for string theory are misleading and maybe the true final physics theory cannot be explained at all to ordinary people, but still it is a nice feature of string theory that it tells a very very nice story.

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1. Yes, strings can be fractals: in fact, smooth strings are the exception. A bit more technically: in the path integral over string worldsheets, smooth worldsheets form a set of measure zero.

2. A good popularization of loop quantum gravity can be found (along with other things) in Lee Smolin’s previous book, Three Roads to Quantum Gravity. There is certainly a ‘nice story’ behind loop quantum gravity, if that’s what one wants — and Smolin tells it pretty well.

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Re:This Week’s Finds in Mathematical Physics (Week 246)

can we have a D2.7-brane?

Tell me what a 3.7-category is and I’ll hand you a 2.7-brane.

More seriously, somebody should remark that John’s comment # just expressed (as Blake certainly knows, but others might not) that we consider physical fields, like the embedding field on the string, usually as elements in some L2L^2-space, rather than in some space of continuous or smooth functions. In particular, fields are often realized in terms of arbitrary Fourier series.

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I don’t know for sure about these fractals. I am just saying that a typical configuration of a string in a background that looks like ℝn\mathbb{R}^n can be thought of as an nn-tuple of real Fourier series on the circle.

This is not supposed to be something special to strings. Similar comments apply to any old QFT, say ϕ4\phi^4-theory, or whatever.

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Thanks, guys! Imagine that physicists had the idea to replace point-particles with Cantor-like sets rather than with strings. Maybe we would have seen a little less geometry and algebra and a bit more analysis in the HEP dish.

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Victor writes:

What is really at stake here is NOT the question of what space and time really really are in the most fundamental sense, but the even more fundamental question of how it is possible to represent them. In other words, we are dealing with not only epistemology but semiotics, the “science” of representation.

Man, I hope this is not considered off-topic. I don’t want to get deleted again. I just want to stress that while Smolin’s thesis, the trouble with physics, is ultimately our inability to unify the discrete and continuous theories, he asserts that there is more than this reflected in the string theory controversy. It’s as if the string controversy is the center piece of the table, focusing our attention on the state of physics as a whole, not just the latest innovation, which might, or might not, have outlived it’s usefulness.

If string theory is justified on mathematical grounds, it’s not just because it is “beautiful mathematics,” but because it’s beautiful mathematics that, to some extent, unifies the discrete and continuous theories. The fact that it has no contact with experiment and can’t predict anything right now, is overridden, in the minds of many, by the apparent achievement of a consistent unification of the discrete and continuous, in a very compelling manner.

The details of how the development of string theory has led to the current prospect of “the end of a science” and consideration of the serious question “What comes next?” are not so important at this point. What is important is gaining a clear understanding of the method of thinking that led us to this point, and without a doubt that thinking is best characterized as the history of developments in the science of mathematics.

String theory must live in a minimum of ten dimensions, but what bothers Glashow, as quoted by Smolin, should bother all of us:

…Worst of all, superstring theory does not follow as a logical consequence of some appealing set of hypotheses about nature. Why, you may ask, do the string theorists insist that space is nine-dimensional? Simply because string theory doesn’t make sense in other kind of space.

In other words, string theory is not inductive science, it’s inventive science, and the comments of Einstein, regarding the significance of epistemology in science, rise to the top of our thoughts, like the cream in unhomogenized milk.

However, and this question must be asked, if string theory is basically an exercise in mathematics, and it is inventive science, then doesn’t that imply that Glashow’s criticism applies to the science of mathematics represented by string theory?

I believe the conclusion that it does is just inescapable. Writing about this in a historical context, Hestenes sees the development of mathematics as the centuries-long effort to unify discrete numbers with continous physcial magnitudes, which Euclid deliberately kept separate, proving theorems first with line segments and then with numbers.

Clearly, though, this history is a history of inventive science, not inductive science. String theory mathematics is simply a continuation to unify discrete numbers with continuous magnitudes. Briefly, the three properties of physical magnitudes versus natural numbers are:

In the development of our inventive mathematical science over centuries, the ad hoc invention of the real numbers addresses number 1; the ad hoc invention of the imaginary numbers addresses number 2; and the ad hoc invention of “compactified dimensions” adresses number 3.

Nevertheless, in the spirit of Glashow’s complaint, wouldn’t we rather have something that “follows as a logical consequence of some appealing set of hypotheses about nature?” But is this even considered possible in mathematics anymore, or is formalism the only “game in town?”

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IMO, the real problem with the anthropic principle is that it is in itself a black hole, leading inexorably to the singularity known as “solipsism.” Once you set foot on that slippery slope you are destined to fall hopelessly into that desolate place from which all possibility of communication with the outside world is lost.

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Have I ever tried to communicate with a solipsist? Heh.

Well, for one thing, a true solipsist would have no reason to communicate with me – or anyone else. That would be pointless, no? Since from that perspective no one else actually exists, only the solipsist.

And yes, it IS a “convenient perspective,” in fact it may well be irrefutable. But there is no way to attempt to communicate such a conviction without immediately contradicting oneself. So while it may be a perfectly logical position it is also indefensible.

Now I have a question for all you physicists out there. If, in the context of the measurement problem, the waveform collapses when a measurement takes place, and a measurement can be meaningful only when it registers in someone’s brain, and the question then is “whose brain,” is it possible to see this situation also as a black hole leading to the same singularity?

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Victor Grauer wrote in part:

a true solipsist would have no reason to communicate with me

This is like saying that a true mathematical formalist (or other sort of mathematical fictionalist or anti-Platonist) would have no reason to think about mathematics. I communicate with you because it is interesting; what other reason should there be?

But there is no way to attempt to communicate such a conviction without immediately contradicting oneself.

I don’t understand why you say this. Have you never mused about philosophy within your own mind? Why shouldn’t a solipsist do the same?

Now I have a question for all you physicists out there.

A very good question! It was thinking about this very question that led me on the path to becoming a solipsist.